Problem: A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$  Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 1)(x - 2)(x + 3).$  Find $r(6).$
Answer: By the Remainder Theorem, $p(1) = -1,$ $p(2) = 3,$ and $p(-3) = 4.$

When $p(x)$ is divided by $(x - 1)(x - 2)(x + 3),$ the remainder is of the form $ax^2 + bx + c.$  Thus,
\[p(x) = (x - 1)(x - 2)(x + 3) q(x) + ax^2 + bx + c\]for some polynomial $q(x).$  Setting $x = 1,$ $x = 2,$ and $x = -3,$ we get
\begin{align*}
a + b + c &= p(1) = -1, \\
4a + 2b + c &= p(2) = 3, \\
9a - 3b + c &= p(-3) = 4.
\end{align*}Subtracting these equations in pairs, we get
\begin{align*}
3a + b &= 4, \\
5a - 5b &= 1.
\end{align*}Solving, we find $a = \frac{21}{20}$ and $b = \frac{17}{20}.$  Then $c = -\frac{29}{10},$ so
\[r(x) = \frac{21}{20} x^2 + \frac{17}{20} x - \frac{29}{10}.\]Thus, $r(6) = \frac{21}{20} \cdot 6^2 + \frac{17}{20} \cdot 6 - \frac{29}{10} = \boxed{40}.$